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A195381 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

%I A195381 #17 Sep 08 2022 08:45:59
%S A195381 1,3,2,3,1,6,9,0,7,6,4,9,9,2,1,4,9,9,5,4,0,3,0,7,3,6,2,4,7,3,5,2,1,7,
%T A195381 4,8,9,9,9,5,4,9,4,0,5,6,1,3,9,5,5,1,0,5,7,5,7,9,8,4,7,1,7,2,2,4,2,3,
%U A195381 1,5,9,5,8,7,8,9,4,2,1,0,7,7,7,2,4,1,5,1,1,8,3,4,1,3,0,7,2,2,0,9
%N A195381 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).
%C A195381 See A195284 for definitions and a general discussion.
%F A195381 Equals sqrt(12)/phi^2, where phi = A001622. - _Jon Maiga_, Nov 14 2018
%e A195381 (A)=1.32316907649921499540307362473521748999...
%t A195381 a = 2; b = Sqrt[5]; c = 3; f = 2 a*b/(a + b + c);
%t A195381 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195381 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195381 x3 = f*Sqrt[2]
%t A195381 N[x1, 100]
%t A195381 RealDigits[%]   (* (A) A195381 *)
%t A195381 N[x2, 100]
%t A195381 RealDigits[%]   (* (B) A195383 *)
%t A195381 N[x3, 100]
%t A195381 RealDigits[%]   (* (C) A195384 *)
%t A195381 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195381 RealDigits[%]   (* Philo(ABC,I) A195385 *)
%t A195381 RealDigits[Sqrt[12] / ((1 + Sqrt[5]) / 2)^2, 10, 100] (* _Vincenzo Librandi_, Nov 15 2018 *)
%o A195381 (Magma) Sqrt(12) / ((1 + Sqrt(5)) / 2)^2; // _Vincenzo Librandi_, Nov 15 2018
%Y A195381 Cf. A195284, A195383, A195384, A195385.
%K A195381 nonn,cons
%O A195381 1,2
%A A195381 _Clark Kimberling_, Sep 17 2011